Colloquium: 2008-2009

FALL 2008

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September 11

Minimal Desingularizations of Planes in Space

Michael WolfRice University

Abstract: We prove that there is only one way, due to Scherk in 1835, to 'desingularize' the intersection of two planes in space and obtain a periodic minimal surface as a result. The proof is mostly an exercise in, and an introduction to, the basic theory of moduli spaces: we translate the geometry of minimal surface in space into a statement about a moduli space of flat structures on Riemann surfaces, and then discuss the deformation theory of and degenerations in this moduli space to prove the result. Naturally, we'll explain all of the terms.

Note: POSTPONED due to Hurricane Ike...reschedule date April 2, 2009

October 2

Bifurcation of Periodic Solutions in Nonlinear Evolution Problems with Periodic Forcing

Patrick RabierUniv of Pittsburgh

Abstract: Click to read abstract »

October 9

Reductive Monoids

Mohan PutchaNorth Carolina State University, Raleigh

Abstract: A reductive monoid M is the Zariski closure of a reductive group G. We will discuss various combinatorial structures associated with their study. First of all the ‘diagonal’ idempotents form the face lattice of a polytope. The second invariant is the cross-section lattice Λ obtained from an idempotent cross-section of the G x G-orbits of M. We illustrate these structures when M is the closure of the images of various natural representations of Mn(k), the multiplicative monoid of all n x n matrices over k. Next the Bruhat-Renner decomposition of M gives rise to a nite monoid R, which for Mn(k), is the rook monoid (=symmetric inverse semigroup), rich in combinatorial and algebraic structure. Finally there is a decomposition of M related to conjugacy classes that gives rise to a nite conjugacy poset C. This combinatorial structure is of yet, very little understood. For the matrix monoid Mn(k), C consists of partitions of m, m ≤ n, ordered by a generalization of the dominance order on partitions of n.

October 16

Workshop on Informatic Phenomena: "Charting the Shape of Quantum State Space"

Christopher FuchsPerimeter Institute for Theoretical Physics, Waterloo, Canada

Abstract: Physicists have become accustomed to the idea that a theory's content is always most transparent when written in coordinate-free language. But sometimes the choice of a good coordinate system can be quite useful for settling deep conceptual issues. This is particularly so for an information-oriented or Bayesian approach to quantum foundations: One good coordinate system may (eventually!) be worth more than a hundred blue-in-the-face arguments. This talk will motivate and chronicle the search for one such class of coordinate systems for finite dimensional operator spaces, the so-called Symmetric Informationally Complete measurements. The desired class will take little more than five minutes to define, but the quest to construct these objects will carry us down a 35 year journey, with the most frenzied activity only recently. If time permits, I will turn the tables and discuss how one might hope to get the formal content of quantum mechanics out of the very existence of such a coordinate system.

October 23

Non-linear Parabolic Flows in Kaehler Geometry

Duong PhongColumbia University

Abstract: A major theme in geometry is to characterize a geometric structure by a "canonical metric", that is, a metric with best curvature properties. A well-known example is the uniformization theorem, which characterizes a complex structure on a surface by a metric with constant scalar curvature. Canonical metrics can be viewed as the fixed points of a suitable flow of metrics. Their existence reduces then to the problem of long-time existence and convergence of such flows. In this talk, we provide a self-contained survey of some of these flows, including the Donaldson heat flow, the Kaehler-Ricci flow, and the Calabi flow. The emphasis will be on the many open problems, and in particular on the conjectures relating the existence of canonical metrics to the algebraic- geometric notion of stability in geometric invariant theory.

October 30

Terrain Data Reconstruction and Approximating PDEs via L1-Minimization

Bojan PopovTexas A&M University

Abstract: In this talk we will consider a class of L1-based minimization methods for solving the following problems:

  1. Digital elevation maps (DEM) for natural and urban terrain
  2. First order Partial Differential Equations (PDEs)
  3. Super-resolution (SR) problems. We will describe each of the three situations. In the case of DEM, various numerical examples will be given. For 1rst order PDEs, a convergence theory will be presented. In the last case, SR problems, we will give a new way to enhance the resolution of digital images. The talk will be accessible to all graduate and advanced undergraduate students.
November 6

Rational Generating Functions and Compositions (joint work with Anders Björner)

Bruce SaganMSU / NSF

Abstract: A composition of the non-negative integer n is a way of writing n as an ordered sum. So the compositions of 3 are 1+1+1, 1+2, 2+1, and 3 itself. It is well-known (and easy to prove) that if c_n is the number of compositions of n then c_n = 2{n-1} for n at least 1 and c_0 = 1. Equivalently, we have the generating function, sum_{n ≧ 0} c_n xⁿ = (1-x)/(1-2x), which is a rational function. We show that this is a special case of a more general family of rational functions associated with compositions. Our techniques include the use of formal languages. Surprisingly, identities from the theory of hypergeometric series are needed to do some of the computations.

November 20

The H-Polynomial of A Group Embedding

Lex RennerUniversity of Western Ontario

Abstract: The Poincare polynomial of a Weyl group calculates the Betti numbers of the projective homogeneous space G/B, while the h-vector of a simple polytope calculates the Betti numbers of the corresponding rationally smooth toric variety. There is a common generalization of these two extremes called the H-polynomial. It applies to projective, homogeneous spaces, toric varieties and, much more generally, to any algebraic variety X where there is a connected, solvable, algebraic group acting with a finite number of orbits. We illustrate this situation by describing the H-polynomials of certain projective G x G -varieties X, where G is a semi-simple group and B is a Borel subgroup of G. This description is made possible by finding an appropriate cellular decomposition for X and then describing the cells combinatorially in terms of the underlying monoid of B x B - orbits. The most familiar example here is the wonderful compactification of a semi-simple group of adjoint type.

December 2

Blocking Light in Closed Manifolds

Ben SchmidtUniversity of Chicago

Abstract: Abstract: To what extent does the collision of light in a closed Riemannian manifold M determine the Riemannian metric on M? I'll discuss conjectures and related results that aim to characterize locally symmetric Riemannian manifolds of non-negative curvature in terms of collisions of light rays.

Note: 2pm GIBSN 126

December 4

Elliptic Curves and Chip-Firing Games on Wheel Graphs

Gregg MusikerMIT

Abstract: In this talk, I will begin by discussing chip-firing games on graphs, and how for a given graph G, this gives rise to a group structure whose order equals the number of spanning trees on G. In the second part, I will describe elliptic curves over finite fields, and how such objects also have group structures. For a family of graphs obtained by deforming the sequence of wheel graphs, the cardinalities of these groups satisfy a nice reciprocal relationship with the orders of elliptic curves as we consider field extensions. I will finish by discussing other surprising ways that these group structures are analogous. Some of this research was completed as part of my dissertation work at the University of California, San Diego under Adriano Garsia's guidance.


January 13

On Knot Homology Theories

Eli GrigsbyColumbia University

Abstract: Understanding knots (smoothly-imbedded circles in 3-manifolds, considered up to isotopy) is essential for understanding 3- and 4-dimensional manifolds. I will discuss two recently-developed tools for studying knots, both inspired by ideas in physics: Khovanov homology and Heegaard Floer homology. In the less than ten years since their introduction, they have generated a flurry of activity and a stunning array of applications. There are also intriguing connections between the two theories that have yet to be fully understood.

Note: 2pm Gibson 310

January 15

Combinatorics of Hopping Particles

Lauren K. WilliamsHarvard University

Abstract: The asymmetric exclusion process (ASEP) is a model from statistical mechanics which describes particles hopping on a one-dimensional lattice. Although it is very simple the model exhibits rich phenomena such as phase transitions. In this talk we will describe surprising connections between the ASEP, the Grassmannian, and orthogonal polynomials. This is joint work with Sylvie Corteel.

January 22

The Topology and Geometry of Heegaard Splittings

Jesse JohnsonYale University

Abstract: A Heegaard splitting is a decomposition of a 3-dimensional topological manifold into simple pieces called handle bodies. The properties of a Heegaard splitting are closely linked to the topology of the 3-manifold, but recent work has shown that they are also intricately related to the geometry of the manifold. I will describe how these connections to geometry have led to insights into the topology of Heegaard splittings.

January 27

On Normalized Ricci Flow and Smooth Structures on 4-Manifolds

Ioana SuvainaCourant Institute of Mathematical Sciences New York University

Abstract: There is a strong relation between the existence of non-singular solutions to the normalized Ricci flow and the underlying smooth structure of the 4-manifold. I am going to discuss an obstruction to the existence of non-singular solutions and its applications. The main examples are connected sums of complex projective planes and complex projective planes with reversed orientation. The key ingredients in our methods are the Seiberg-Witten Theory and symplectic topology. This is joint work with M. Ishida and R. Rasdeaconu.

Note: 2pm

January 29

Mathematical Modeling of Cell-Matrix Interactions in Articular Cartilage

Mansoor HaiderNorth Carolina State University

Abstract: Articular cartilage is the primary load bearing soft tissue in diarthrodial joints such as the knee, shoulder and hip. Cartilage is comprised of a solid phase extracellular matrix that is saturated by an interstitial fluid phase. Homeostasis in the tissue is maintained by a single population of cells called chondrocytes. Since cartilage contains no blood vessels or nerve endings, the local environment of the chondrocytes is known to strongly influence cellular metabolic activities. In this talk, I will present mathematical models of biomechanical and biophysical interactions between the cells of articular cartilage and their local environment. I will also discuss relevance of the models to understanding the role of physical factors in osteoarthritis, and to the development of tissue engineering strategies for cartilage repair or regeneration using hydrogels.

February 5

On Topological Bounds for the Energy of Fluid Flows

Rafal KomendarczykUniversity of Pennsylvania

Abstract: In many physical situations we seek lower bounds for the L²-energy of a volume preserving vector field in terms of topological invariants of the field. A typical example is evolution of a magnetic field in ideal magnetohydrodynamics or a vorticity field in hydrodynamics, governed by the Euler equations.

Classical energy bounds obtained by V.I. Arnold or M. Freedman and Z.-X. He, involve invariants, such as Woltjer's helicity, which measure linkage of orbits in the flow.

I will present a new perspective on these invariants, which exploits connections between homotopy theory of maps associated to n-component links and classical link-homotopy invariants discovered by Milnor in his senior thesis.

As a particular application, I will derive a new lower bound for the energy of a volume preserving vector field in terms of a helicity measuring the third order linkage of orbits.

Note: 2pm Gibson 310

February 5

Computational Investigations of Intrinsic and Extrinsic Mechanisms Underlying Formation of Carcinomas

Kasia A. RejniakMoffitt Cancer Center & Research Institute Integrative Mathematical Oncology

Abstract: A disruption of epithelial morphogenesis is thought to be involved in the initiation of cancer, however, little is known about the cell biology of early neoplastic lesions. Computational models can facilitate in the study of early cancer lesions by directing essential empirical data collection and by integrating results in quantitative outcomes. Here, we present IBCell, a computational model of epithelial acinar structures that predicts the disruptive effects of altered cell-microenvironment interactions on epithelial morphogenesis. A systematic investigation using IBCell reveals a range of model parameters (in terms of cell sensitivity to external cues) for which robust acinar structures form, and the ranges of parameters leading to abnormal geometrical forms resembling tumor-like morphologies, such as: acini with filled lumen or cell clusters with uncontrolled growth. The results obtained from computational simulations are compared (in both, a qualitative and quantitative way) to three-dimensional in vitro experiments on a non-tumorigenic MCF10A cells and some mutants derived from this cell line.

February 12

Hirzebruch Invariants of Complex Algebraic Varieties

Laurentiu MaximCUNY

Abstract: Abstract: An old theorem of Chern, Hirzebruch and Serre asserts that the signature of closed oriented manifolds is multiplicative in fiber bundles with trivial monodromy action (i.e., bundles for which the fundamental group of the base acts trivially on the cohomology of the fiber). The contribution of monodromy to the signature of a fiber bundle was later described by Atiyah and Meyer. In this talk I will survey various extensions of these results to the singular setting, and discuss parametrized versions of them in the complex algebraic context. The talk will be suitable for a general audience.

Note: 2pm Gibson 310

February 12

An Efficient Algorithm for Shape Optimization on Elliptic Eigenvalue Problems

Chiu-Yen KaoOhio State University

Abstract: In this talk, we will discuss an efficient numerical approach to find the optimal shape and topology for elliptic eigenvalue problems in an inhomogeneous media. The method is based on Rayleigh quotient formulation of eigenvalue and a monotone iteration process to achieve the optimality. The common numerical approach for these problems is to start with an initial guess for the shape and then gradually evolve it, until it morphs into the optimal shape. One of the difficulties is that the topology of the optimal shape is unknown. Developing numerical techniques which can automatically handle topology changes becomes essential for shape and topology optimization problems. The level set approach based on both shape derivatives and topological derivatives has been well known for its ability to handle topology changes. However, CFL constrain significantly slows down the algorithm when the mesh is further refined. Due to the efficient binary update, our method not only has the ability of topological changes but also is exempt from CFL condition. We provide numerous numerical examples to demonstrate the robustness and efficiency of our approach.

February 19

What Is a Syzygy?

Adam Van TuylLakehead University

Abstract: In this talk I will provide an elementary introduction to the idea of a syzygy. A syzygy, as I shall explain, is an abstraction of the null space of a matrix. I will also explore some connections between syzygies and graph theory. This talk assumes only a background in undergraduate abstract algebra and linear algebra.

February 26

Rook Polynomials in Combinatorics

Jim HaglundUniversity of Pennsylvania

Abstract: We survey various topics in rook theory, starting with how to count permutations with restricted position, then covering theorems and conjectures on the zeros of rook and matching polynomials, and finally highlighting applications to permutation statistics.

March 12

Parking Functions, Trees, and Diagonal Harmonics

Nick LoehrVirginia Tech

Abstract: Parking functions are combinatorial objects that have connections to hashing theory, the enumeration of trees, representation theory, and algebraic combinatorics. This talk describes the basic properties of parking functions and their quantum analogues. In particular, we describe a conjecture (due to J. Haglund and the speaker) that expresses the Hilbert series for the space of diagonal harmonics as a sum of suitably weighted parking functions.

April 2

Minimal Desingularizations of Planes in Space

Michael WolfRice University

Abstract: We prove that there is only one way, due to Scherk in 1835, to 'desingularize' the intersection of two planes in space and obtain a periodic minimal surface as a result. The proof is mostly an exercise in, and an introduction to, the basic theory of moduli spaces: we translate the geometry of minimal surface in space into a statement about a moduli space of flat structures on Riemann surfaces, and then discuss the deformation theory of and degenerations in this moduli space to prove the result. Naturally, we'll explain all of the terms.

April 9

A Real Variable Method and Its Applications

Zhongwei ShenUniversity of Kentucky

Abstract: In this talk we will describe a real variable method and its applications to problems in analysis and partial differential equations. Starting with L² estimates, this method allows us to establish Lp estimates via certain scale-invariant weak reverse Holder inequalities.

April 16

A Multi-Resolution Method for Climate System Modeling: Application of Spherical Centroidal Voronoi Tessellations

Max GunzbergerFlorida State University

Abstract: During the next decade and beyond, climate system models will be challenged to resolve scales and processes that are far beyond their current scope. Each climate system component has its prototypical example of an unresolved process that may strongly influence the global climate system, ranging from eddy activity within ocean models, to ice streams within ice sheet models, to surface hydrological processes within land system models, to cloud processes within atmosphere models. These new demands will almost certainly result in the develop of multi-resolution schemes that are able, at least regional, to faithfully simulate these fine-scale processes. Spherical centroidal Voronoi tessellations (SCVTs) offer one potential path toward the development of robust, multi-resolution climate system component models. SCVTs allow for the generation of high quality Voronoi diagrams and Delaunay triangulations through the use of an intuitive, user-defined density function. In each of the examples provided, this method results in high-quality meshes where the quality measures are guaranteed to improve as the number of nodes is increased. Real-world examples are developed for the Greenland ice sheet and the North Atlantic ocean. Idealized examples are developed for ocean-ice shelf interaction and for regional atmospheric modeling. In addition to defining, developing and exhibiting SCVTs, we pair this mesh generation technique with a previously developed finite-volume method. Our numerical example is based on the nonlinear shallow-water equations spanning the entire surface of the sphere. This example is used to elucidate both the potential benefits of this multi-resolution method and the challenges ahead.

This is joint work with Lili Ju (University of South Carolina) and Todd Ringler (Los Alamos National Laboratory).

Note: 2pm Gibson 310

April 23

2-Dimensional Sloshing Flows for the Shallow Water Equations

James GreenbergCarnegie Mellon University

Abstract: In this talk I'll discuss "sloshing" 2-Dimensional flows for the "shallow-water" equations. The model describes the motion of a finite volume of viscous fluid taking place in container whose bottom is described by a paraboloidal-like surface of the form:

z = (αx² + βy²) / 2, α > 0, β > 0

or more generally

z = a (x, y )

where a → ∞ as (x² + y² ) → ∞. The model includes gravity, coriolis, and viscous forces.

Mathematics Department, 424 Gibson Hall, New Orleans, LA 70118 504-865-5727